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please explain each step

YEEEEEET  Nov 21, 2018
edited by YEEEEEET  Nov 21, 2018
 #1
avatar+92429 
+1

13^n    -  6^(n - 2)

 

Show it's true for n = 2

 

13^2 - 6^(2 - 2)   =

169 - 1  =  168

168 / 7   =  24

 

Assume that this is true for  n = k     where k ≥ 2

That is

13^k - 6^(k - 2)      is divisible by 7

 

Prove it's true for k + 1

That is

13^(k + 1) -  6^ (k + 1 - 2)   is divisible by 7       

 

[ note ....6^(k + 1 - 2)  = 6^(k - 2 + 1) ]

 

 

So we have 

 

13^(k+ 1)  - 6 ^( k - 2 + 1)

 

13^k * 13^1  -  6^(k-2) * 6^1

 

13 * 13^k -   6 * 6^(k - 2)

 

(6 + 7) 13^k - 6 * 6^(k - 2)

 

6 [ 13^k - 6^(k - 2) ]   + 7 * 13^k

 

And since we assumed that 13^k - 6^(k - 2) was divisible by 7, then the first term is divisible by 7, as well

 

And  the second term, 7*13^k,    is divisible by 7

 

 

cool cool cool

CPhill  Nov 21, 2018

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